cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A368332 The number of terms of A054744 that divide n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

The number of divisors d of n such that e >= p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
The largest of these divisors is A368333(n).

Links

Crossrefs

Cf. A054744, A365632, A368328, A368333, A368334, A368335.

Programs

  • Mathematica
    f[p_, e_] := If[e < p, 1, e - p + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] < f[i,1], 1, f[i,2] - f[i,1] + 2));}

Formula

Multiplicative with a(p^e) = 1 if e < p, and a(p^e) = e - p + 2 if e >= p.
a(n) >= 1, with equality if and only if n is in A048103.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^(p*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^(p-1))) = 1.58396891058853238595... .

A368328 The number of terms of A054743 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Comments

The number of divisors d of n such that e > p for all prime powers p^e in the prime factorization of d (i.e., e >= 1 and p^(e+1) does not divide d).
The largest of these divisors is A368329(n).

Links

Crossrefs

Cf. A054743, A207481, A365632, A368329, A368330, A368331, A368332.

Programs

  • Mathematica
    f[p_, e_] := If[e <= p, 1, e - p + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] <= f[i,1], 1, f[i,2] - f[i,1] + 1));}

Formula

Multiplicative with a(p^e) = 1 if e <= p, and a(p^e) = e - p + 1 if e > p.
a(n) >= 1, with equality if and only if n is in A207481.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 1/p^((p+1)*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/((p-1)*p^p)) = 1.27325025767774256043... .

A365634 The number of divisors of n that are terms of A048102.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Sep 14 2023

Keywords

Links

Crossrefs

Cf. A048102, A365632, A365635.

Programs

  • Mathematica
    f[p_, e_] := If[e == p, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == f[i,1], 2, 1));}

Formula

Multiplicative with a(p^e) = 1 + [e = p], where [] is the Iverson bracket.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/p^(p+1)) = 1.153074089009... .

A368336 The number of divisors of the largest term of A072873 that divides of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 5, 4, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Dec 21 2023

Keywords

Links

Crossrefs

Cf. A000005, A048103, A072873, A327939, A365632, A368331, A368335.

Programs

  • Mathematica
    f[p_, e_] := (e - Mod[e, p] + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,2] - f[i,2]%f[i,1] + 1);}

Formula

a(n) = A000005(A327939(n)).
Multiplicative with a(p^e) = e - (e mod p) + 1.
a(n) >= 1, with equality if and only if n is in A048103.
a(n) <= A000005(n), with equality if and only if n is in A072873.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + p/(p^p-1)) = 1.86196549645040699446... .

A365633 The sum of divisors of n that are terms of A072873.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Amiram Eldar, Sep 14 2023

Keywords

Comments

The number of these divisors is A365632(n) and the largest of them is A327939(n).

Links

Crossrefs

Cf. A072873, A327939, A332653, A365632.

Programs

  • Mathematica
    f[p_, e_] := (p^(Floor[e/p] + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(1+f[i,2] \ f[i,1])-1)/(f[i,1] - 1));}

Formula

Multiplicative with a(p^e) = (p^(floor(e/p)+1) - 1)/(p - 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (A332653(p)/(p^(p-1)-1) - 1/(p*(p-1))) = 2.253624924813... .
Showing 1-5 of 5 results.

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